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Essential Math for Games: Constrained Dynamics and Simulations

Learn about various types of constraints and their applications in game development, including box constraints, bead on a wire, spring constraints, cloth simulation, articulated bodies, and angular constraints. Understand how to solve these constraints using linear systems.

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Essential Math for Games: Constrained Dynamics and Simulations

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  1. Constrained Dynamics Marq Singer (marq@essentialmath.com)

  2. The Problem • What are they • Why do we care • What are they good for Essential Math for Games

  3. The Basics • Constraint – something that keeps an entity in the system from moving freely • For our purposes, we will treat each discreet entity as one particle in a system • Particles can be doors on hinges, bones in a skeleton, points on a piece of cloth, etc. Essential Math for Games

  4. Box Constraints • Simplest case • Movement constrained within a 2D area Essential Math for Games

  5. Box Constraints P 0 100 Essential Math for Games

  6. Box Constraints (cont) • Restrict P to extents of the box • Recover from violations in position (last valid, rebound, wrap around) • Simple, yet the basis for the rest of this Essential Math for Games

  7. Bead on a Wire • The Problem: • Restrict bead to path • Solutions: • Explicit (parametric) method • Implicit method Essential Math for Games

  8. Parametric Constraints Essential Math for Games

  9. Bead on a Wire • From Baraff, Witkin N = gradient f = force fc = constraint force f‘ = f + fc Essential Math for Games

  10. Implicit Representation legal position legal velocity legal acceleration Essential Math for Games

  11. Implicit Representation Constraint force = gradient vector times scalar l Essential Math for Games

  12. Spring Constraints • Seems like a reasonable choice for soft body dynamics (cloth) • In practice, not very useful • Unstable, quickly explodes Essential Math for Games

  13. Stiff Constraints • A special spring case does work • Ball and Stick/Tinkertoy • Particles stay a fixed distance apart • Basically an infinitely stiff spring • Simple • Not as prone to explode Essential Math for Games

  14. Cloth Simulation • Use stiff springs • Solving constraints by relaxation • Solve with a linear system Essential Math for Games

  15. Cloth Simulation Essential Math for Games

  16. Cloth Simulation • Forces on our cloth Essential Math for Games

  17. Cloth Simulation • Relaxation is simple • Infinitely rigid springs are stable • Predetermine Ci distance between particles • Apply forces (once per timestep) • Calculate D for two particles • If D != 0, move each particle half the distance • If n = 2, you’re done! Essential Math for Games

  18. Relaxation Methods Essential Math for Games

  19. Relaxation Methods Essential Math for Games

  20. Cloth Simulation • When n > 2, each particle’s movement influenced by multiple particles • Satisfying one constraint can invalidate another • Multiple iterations stabilize system converging to approximate constraints • Forces applied before iterations • Fixed timestep (critical) Essential Math for Games

  21. More Cloth Simulation • Use less rigid constraints • Vary the constraints in each direction (i.e. horizontal stronger than vertical) • Warp and weft constraints Essential Math for Games

  22. Still More Cloth Simulation • Sheer Springs Essential Math for Games

  23. Still More Cloth Simulation • Flex Springs Essential Math for Games

  24. Articulated Bodies • Pin Joints • Hinges Essential Math for Games

  25. Angular Constraints • Restrict the angle between particles • Results in a cone-shaped constraint Essential Math for Games

  26. Angular Constraints • Unilateral distance constraint • Only apply constraint in one direction Essential Math for Games

  27. Angular Constraints • Dot product constraint • Recovery is a bit more involved Essential Math for Games

  28. Stick Man • Uses points and hinges • Angular (not shown) allow realistic orientations • Graphic example of why I’m an engineer and not an artist Essential Math for Games

  29. Using A Linear System • Can sum up forces and constraints • Represent as system of linear equations • Solve using matrix methods Essential Math for Games

  30. Basic Stuff Systems of linear equations Where: A = matrix of coefficients x = column vector of variables b = column vector of solutions Essential Math for Games

  31. Basic Stuff • Populating matricies is a bit tricky, see [Boxerman] for a good example Isolating the ith equation: Essential Math for Games

  32. Jacobi Iteration Solve for xi (assume other entries in x unchanged): (Which is basically what we did a few slides back) Essential Math for Games

  33. Jacobi Iteration In matrix form: D, -L, -U are subparts of A D = diagonal -L = strictly lower triangular -U = strictly upper triangular Essential Math for Games

  34. Jacobi Iteration Definition (diagonal, strictly lower, strictly upper): A = D - L - U Essential Math for Games

  35. Gauss-Seidel Iteration Uses previous results as they are available Essential Math for Games

  36. Gauss-Seidel Iteration In matrix form: Essential Math for Games

  37. Gauss-Seidel Iteration • Components depend on previously computed components • Cannot solve simultaneously (unlike Jacobi) • Order dependant • If order changes the components of new iterates change Essential Math for Games

  38. Successive Over Relaxation (SOR) • Gauss-Seidel has convergence problems • SOR is a modification of Gauss-Seidel • Add a parameter to wG-S Essential Math for Games

  39. Successive Over Relaxation (SOR) • = a Gauss-Seidel iterate • 0 < w < 2 • If w = 1, simplifies to plain old Gauss-Seidel Essential Math for Games

  40. Gauss-Seidel Iteration In matrix form: Essential Math for Games

  41. Lots More Math(not covered here) • I highly recommend [Shewchuk 1994] • Steepest Descent • Conjugate Gradient • Newton’s Method (in some cases) • Hessian • Newton variants (Discreet, Quasi, Truncated) Essential Math for Games

  42. References • Boxerman, Eddy and Ascher, Uri, Decomposing Cloth, Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2004) • Eberly, David, Game Physics, Morgan Kaufmann, 2003. • Jakobsen, Thomas, Advanced Character Physics, Gamasutra Game Physics Resource Guide • Mathews, John H. and Fink, Kurtis K., Numerical Methods Using Matlab, 4th Edition, Prentice-Hall 2004 • Shewchuk, Jonathan Richard, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain, August 1994. http://www-2.cs.cmu.edu/~jrs/jrspapers.html • Witken, Andrew, David Baraff, Michael Kass, SIGGRAPH Course Notes, Physically Based Modeling, SIGGRAPH 2002. • Yu, David, The Physics That Brought Cel Damage to Life: A Case Study, GDC 2002 Essential Math for Games

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